We showed that the conventional expansion
in powers of the field for
the critical potential
of 3-dimensional O(N) models in the
large-N limit, does not converge for values of
larger than some critical value.
We found numerical evidence for
conjugated branch points in the complex plane (see figure below).
Padé approximants [L+3/L] for the critical potential
apparently converge at large
.
This allows high-precision
calculation of the fixed point in a more
suitable set of coordinates.
We argue that the singularities are
generic and not an artifact of the large-N limit.
Ignoring these singularities
may lead to inaccurate approximations.
Figure 9: Real and imaginary parts of the roots of the denominator
(filled squares) and numerator (crosses) of
a [26/23] Padé approximant for the critical potential.
More details in:
Y. Meurice, Complex Singularities of the Critical Potential in the
Large-N limit, hep-th/0208181, Phys. Rev. D (in press).
Work in Progress, Plans:
We have studied the critical properties (fixed points, exponents) of O(N)
hierarchical non linear sigma models at values of N ranging from 1
to 130. This work involved J. J. Godina, B. Oktay and L. Li. We are
presently attempting to determine the first four coefficients of the
1/N expansion in order to get some idea about the Borel summability
of the series. Preliminary results seem to favor the Borel summability.
We have observed that in a system of coordinates where
the unstable fixed point can be approximated by polynomials, the procedure
which consists in considering bare potential truncated at order
has a
low accuracy.
We are planning to investigate if similar problems
appear near tricritical fixed points. In particular,
reconsidering the RG flows in
a larger space of bare parameters may affect the generic
dimension of
the intersections of hypersurface of various codimensions and help us
finding a more general realization of spontaneous breaking of
scale invariance with a
dynamical generation of mass as in the Bardeen-Moshe-Bander mechanism.